Communications inCommun. Math. Phys.72, 277-292 (1980) Mathematical

Physics© by Springer-Verlag 1980

Arbitrary N-Vortex Solutionsto the First Order Ginzburg-Landau Equations*

Clifford Henry Taubes

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Abstract. We prove that a set of N not necessarily distinct points in the planedetermine a unique, real analytic solution to the first order Ginzburg-Landauequations with vortex number N. This solution has the property that the Higgsfield vanishes only at the points in the set and the order of vanishing at a givenpoint is determined by the multiplicity of that point in the set. We prove furtherthat these are the only C°° solutions to the first order Ginzburg-Landauequations.

1. Introduction

A mathematical model of superconductors is given by the Ginzburg-Landauequations [1]. These equations exhibit vortex solutions which may be viewed asfinite energy solutions to the equations describing the two dimensional AbelianHiggs model [2]. After suitable rescaling, the equations have one couplingconstant, λ, whose value determines whether the equations describe a type I or typeII superconductor. The value λ = 1 is the critical value. In this case, the energy of aconfiguration is bounded below by a topological invariant, a multiple of the vortexnumber. Any configuration which achieves this minimum energy will be a solutionof the Ginzburg-Landau equations. To find solutions with this minimum bound,one need only solve a set of first order coupled equations for the vector potentialand the Higgs field rather than the more general second order equations. DeVegaand Schnaposnik [3] in an analysis of the equations, gave a numerical argumentfor the existence of a cylindrically symmetric solution to the first order equationswith vortex number N which has been interpreted as N-vortices superimposed at apoint. Weinberg [4] recently proved that if a solution of the Ginzburg-Landauequations with vortex number N exists then the dimension of the space of moduliof this solution must be 2N. This led him to conjecture that there exists a 2AΓparameter family of solutions with vortex number TV and that the parameters of asolution may be interpreted as being the positions in R2 of the N vortices. Recent

* This work is supported in part through funds provided under Contract PHY 77-18762

278 C. H. Taubes

numerical work by Jacobs and Rebbi [5] concerning the interaction energy of apair of vortices lends support to this conjecture.

In this paper we prove the validity of Weinberg's conjecture. To be precise, weprove that the solutions to the first order Ginzburg-Landau equations withcoupling constant λ—\ are uniquely determined by a set of N not necessarilydistinct points in the plane corresponding to the zeros of the Higgs field. Every setof AT points determines one such solution. The vortex number of this solution is N.The solution manifold of the first order Ginzburg-Landau equations with vortexnumber N is isomorphic to IR2N.

II. The Equations

After a suitable rescaling, the action for the two-dimensional Abelian Higgs modelis:

sί= \ d2xl(δJ-iAJ)φl2 + ±FjkFJk + (\φ\2-ί}2 . (2.1)R2 I δ J

The Higgs field φ is a complex scalar field, or alternatively it may beinterpreted as a cross-section of a complex line bundle over IR2. The A are thecomponents of the connection on the line bundle; Fjk = djAk — dkAj is thecurvature. The critical value of the coupling constant A is 1. Henceforth we shallonly discuss this case. Finite action requires that a configuration (AJ9 φ) have thefollowing behavior as |x| approaches infinity:

M->1(2.2)

DAφ = dφ — iAφ-^ Q, as |x|-»oo.

The usual ίopological arguments imply that the vortex number,

n=±- Jd 2xF 1 2 (2.3)In R2

is an integer and is unchanged by finite action perturbations of the fields. Theinteger n is a topological invariant the first Chern number of the complex linebundle in which A is a connection.

As BogomoΓnyi [6] pointed out, a lower bound on the action results fromrewriting the action via an integration by parts as

±i ί d2xF12 . (2.4)ΊR2

The upper sign corresponds to positive vortex number and the lower sign tonegative vortex number. In Eq. (2.4) φί = Reφ and φ2 = lmφ. We shall treat thecase of positive vortex number only, the case of n<0 being completely analogous.

Vortex Solutions to the Landau-Ginzburg Equations 279

The lower bound of BogomoΓnyi is

(2.5)

This lower bound is realized if and only if

2-A2φί) = 0, (2.6a)

2-A1φ1) = 09 (2.6b)

= V. (2.6c)

Equations similar to these arise in the study of cylindrically symmetric self-dualsolutions to four dimensional SU(2) Yang-Mills equations [7]. As in that case, it ispossible to reduce Eqs. (2.6) to one nonlinear second order differential equation forone unknown function. To do this set

. (2.7a)

Equations (2.6a) and (2.6b) are the real and imaginary parts of the complexvalued equation

2dφ-ίAφ = 0 (2.7b)

This equation may be solved for A

A=-2id\nφ. (2.8)

Define the complex valued function /=/i + if2 with f± and /2 real valued on IR2 by

φ = ef. (2.9)

In terms of the real functions /x and /2 the components of A and the curvature are

(2.10)

The asymptotic condition that \φ\ = ί as |x| approaches infinity demands that

Lim/1(x) = 0. (2.|x|-» oo

Equation (2.6c) is a second order nonlinear differential equation for /x [5] :

(2.12)

We reinterpret the topological invariant n as defined in equation (2.3) in termsof the functions f± and /2. For large values of |x|,

φ-+eίf2, as |x|-»oo (2.13a)

and

, (2.13b)

280 C. H. Taubes

where n is the vortex number defined in Eq. (2.3). Homotopy considerations implythat no continuous function on IR2 can have this asymptotic behavior. TheGinzburg-Landau equations leave /2 undetermined. This is the manifestation ofthe gauge in variance of the original lagrangian. Any /2 with the above asymptoticbehavior must be singular on some set in the plane. If we are interested in at leastcontinuous solutions to the first order Ginzburg-Landau equations, then φ mustvanish on a nontrivial set in the plane and there must exist a function /2 satis-fying Eq. (2.13b) which has its singularities only on this set. In the Appendix weshall prove that a necessary condition for φ to solve the first order Ginzburg-Landau equations is that the zero set of φ be discrete. For now we shall assumethat this is the case. If we demand that our solutions be C1 on IR2, we are furtherrestricted to have φ vanish as an integer power of (x — at) as x approaches at for a{ azero of φ. Any solution to the first order equations determines a set of N points(αl5 ...,flN) in IR2. These points need not be distinct, the number of times a givenpoint occurs corresponds to the order of vanishing of φ at that point. As x-^ak thefunction /x must have the behavior:

;-+^ln(x-αk)2, (2.14)

\x\-+ak 2

where nk is the order of vanishing of φ at ak. Given /x with this behavior, we canfind a function /2 with the asymptotic behavior expressed in (2.13b) which is in fact

C » i n R 2 \ U K } :\*=1

Λ= Σ 0*. (2.15)k = l

where

k2

The topological charge, n, is equal to the size, JV, of the set (α1,α2, ...,<%).A simple calculation shows that with /2 given above and fl as in Eq.(2.14),

\ n

vanishing as |x|-*oo and C^inRΛ (J {ak} the components of the connection\ k = l

[Eq.(2.10)] are in fact infinitely differentiable on ]R2.

III. The Results

Up to now, solutions to the first order Ginzburg-Landau equations with vortexnumber n are thought to exist only when the n points (α1 ?..., an) are the same [3].The main result of this paper is the following theorem:

Theorem 1. To each point (α1?...,αj in IR2 x ... xIR2 = IR2", there exists a uniqueglobally C°° solution to Eqs. (2.6a)-(2.6c) which satisfies the conditions of Eqs.(2.2)and (2.3) and satisfies the further conditions that

{xeIR2|0(x)=0}= [J {αj (3.1)

Vortex Solutions to the Landau-Ginzburg Equations 281

with the order of vanishing of φ at a point α0 being precisely the number of times a0

appears in the set {al9...>an}. ΠFrom the discussion of the last section, we will have proved Theorem 1 if we

can prove that Eq.(2.12) has a unique solution, f ί 9 such that /Ί is C°° in\ n

EM (J {akϊ ' Eq.(2.11) is satisfied as |x|->oo and as x approaches each point ak,f^\ f c = l

has the behavior described by Eq.(2.14). To do this, let

(3.2)

with λ>4n a real number. On IR2\ \J {ak}, u0 is infinitely differentiate andformally \ f e = 1

+4π Σ *(*-**)• (3-3)

Here, <5(*) is ίhe Dirac delta function. Define a function g0 by

so that on IR2\ (J {αfc}, g0 and -Jw0 agree. Next we define a new unknown\ k = l

function υ by :

2fl=u0 + v. (3.5)

If /j satisfies Eq.(2.12) and has the behavior at infinity and at the points {αk}discussed above, then v is a solution to the equation

:0, (3.6a)

lim t; = 0. (3.6b)|x|->oo

Theorem I then follows from

Theorem I'. For every point (αl9 . . .,αn) in IR2 x IR2 x . . . x IR2 =1R2" and u0 and g0

defined by Eqs.(3.2) and (3.4), respectively, there exists a unique function v, realanalytic in IR2, satisfying (3.6a) and (3.6b). Π

We observe that (3.6a) is formally the variational equations of the functional

φ) = J d2x&μvdμv -~v(l-g0) + e»°(ev - 1)) . (3.7)IR2

The strategy for proving Theorem ΐ is to define the functional α(υ) on anappropriate Banach space and then appeal to the following results of functionalanalysis. For a detailed exposition of the techniques used here see e.g.,Vainberg [8].

Proposition 3.1. If α strictly convex functional G(x) defined in a linear space E has aminimum at a point x0, then x0 is an absolute minimum point, and there are no otherminimum points. Π

282 C. H. Taubes

For a proof, see [8], p. 96.In Sect. IV we show that there exists an appropriate Banach space on which

a(v) is strictly convex. This proves the uniqueness assertion of Theorem Γ.The existence of a solution will follow if we can show that a(v) satisfies the

hypothesis of

Proposition 3.2. Let G(x) be a real Gateaux differentiable functional defined in a realreflexive Banach space E, which is weakly lower semi-continuous and satisfies thecondition

<G'(x),x>>0 (3.8)

for any vector xe£, ||x|| =β>0, and Gf(x) = grad G(x). Then there exists an interiorpoint x0 of the ball \\x\\ ^ Rat which f(x) has a local minimum so that G'(x) = 0. Π

For a proof, see [8], p. 100.We prove in Sect. V that a(υ) satisfies the hypothesis of Proposition 3.2. This

proves the existence of weak solution to Eq.(3.6a).In Sect. VI we prove that the solution must in fact be C00 on 1R2.

IV. Properties of a(v)

We begin by defining a Banach space H to be the completion of C^(1R2) (the spaceof infinitely differentiable functions with compact support) in the norm

f fLiR2

(4.1)

The space H is the first Sobolev space on R2. If Ω is any Lebesgue measurableset in IR2, define for 1 p, q < oo

2 \ Ί l / p

Σμ=l

(4.2)

The functional a(v) as defined in Eq. (3.7) is defined on C^(IR2) which is densein£Γ.

We state the fundamental properties of the functional a(v) in the following setof propositions and their corollaries :

Proposition 4.1. Define the functional a(v) on C^ by

)= ί &8μυdμv-v(l -00) + έ? V- 1)} (4.3)

with u0 and g0 defined by Eqs. (3.2) and (3.4); then a(v) extends to a nonlinearfunctional on H with domain H. Π

Proposition 4.2. The Gateaux derivative of a(v\ a'(v,h), exists for all v, heH and

af(v,h)=lim-(a(v + th)-a(v))= J {duvdh-h(l-g0-eu°} + heu°(ev-l)} . (4.4)ί^O t i

Vortex Solutions to the Landau-Ginzburg Equations 283

Furthermore, for fixed v, a'(v, •) is a bounded linear functional on H, denoted(grada(v\hy. For fixed heH, a'( ,h) is a nonlinear functional on H with domainH. Π

Proposition 4.3. The functional a(v) is strictly convex on H. Π

Before proving these propositions we state and prove some immediatecorollaries:

Corollary 4.4. The functional a(υ) is weakly lower semi-continuous on H. Π

Proof of Corollary 4.4. From Proposition 4.2 the Gateaux differential a'(v, ) forfixed veH is bounded and hence continuous on H. The result follows fromTheorem8,6 of Vainberg[8], p.82.

Corollary 4.5. v0 is a minimum of the functional a(v) if and only if grad a(v0) — 0. Π

Proof of Corollary 4.5. From Proposition 4.2 Gateaux differential, a'(v, •) for fixed vis linear. From Proposition 4.3, a(v) is convex on H. The result follows fromTheorem 9.1 of Vainberg[8], p. 91.

We remark that Proposition 4.3 implies that if a(v) has a minimum on H, thatminimum is unique. Corollary 4.5 states that if the minimum exists, then it is aweak solution to Eqs.(3.6a) and (3.6b).

Proof of Proposition 4.1, We must show that if ve H then a(v) is finite. Write a(v) asa sum of three terms:

φ) = j $dμυdμυ- f v(ΐ-g0-O + f <*>(ev-l-v). (4.5)R2 R2 R2

We note that the first term on the right in Eq. (4.5) is bounded by l/2| |z;| |I 2. UsingHolder's inequality we have for the second term

Since 1 — gΌ — eu° is in L2(ΊR2, d2x) the second term is bounded on H. For the thirdterm in Eq.(4.5) we have

J euo(ev-1-v) <: $ \ev-ί-v\. (4.7)[R2 R2

It remains to be shown that the right hand side of Eq.(4.7) is finite for all veH.For ρ>0, define I^QV) by

I1(ρv)= J (exp(ρ2ι;2)— i}dx. (4.8)R2

Given veH, there exists ρ>0 such that I1(ρv)<ao (cf. [9], pp.242-246). Fix

such a ρ. To estimate J \ev— 1 — v\ define sets ί21? Ω2 ClR2 byR2

,! ' (4-9)

284

In Ωj we have ρ2v2^.v, hence

In ί22, f is bounded. Hence there exists a constant c > 0 such that

in Ω2 .

Therefore :

C. H. Taubes

(4.10)

(4.11)

(4.12)J \eυ-l-v\^I2 + c J t;2<oo ,1R2 R2

and we have proved

Lemma 4.6. For α// t eίί, the functional

R(v)= j |eϋ-l-ί;|R2

is finite.

Proof of Proposition 4.2. Let v, heH. The Gateaux derivative of a(v) is bydefinition

„ M Γ αα (t; ft) = hm

-

=}!ίs{f1 ° U2

= J {dμvc

(4.13)

To prove Proposition 4.2 we must first show that Eq. (4.4) is correct. Once we haveestablished Eq. (4.4) it remains for us to demonstrate that for fixed ueH, a'(υ, •) is abounded functional on H and for fixed heH, the nonlinear functional 0'( ,ft) hasdomain H.

Lemma 4.7. For any heH

Proof of Lemma 4.7. We first note that

(4.14)

(4.15)

For the remaining term in Eq. (4.14) we have the bound

lim I*(eth-

°ev

R2

'Ί(eth —

Q

1

ί

1

ί

— ίft)

-ίΛ)

< lim f~ ί-O R2

<«•-!)

^uo(eth-l-th)1

(4.16)

Vortex Solutions to the Landau-Ginzburg Equations 285

In order to show that the right hand side of (4.16) is zero in the limit that t goesto zero we need the estimate

Lemma 4.8. For any veH and k^2

II k \

Proof of Lemma 4.8. This estimate is proved in Adams [9], pp. 103-106.

We return to the proof of Lemma 4.7. For the first term on the right hand sideof (4.16) we have

1 I i^.i.^l i i^-i-ίftig I f; lί*-W (4.18)1 R2 l R2 R2 fc=2 K

In (4.18) we have defined eth by its power series. By the Monotone ConvergenceTheorem we can interchange the order of integration and summation. Using (4.17)we have

J \e«°(eth-i-th)\^4 t-i (J/2||ι;||1>2;R2)fe. (4.19)

fc=2

Next we remark that by Stirling's formula, n" is asymptotic to nlen(2Πn)l/2 andthat 3iV>0 such that

(4.20)

Choose N such that (4.20) holds. Then

1 J (^-1-^)1^4 £ ί""1 ^(V^INl.***)*ί ro2 J.-J Kl

+ 4 Σ ί*-H]Λ2φ|lι,2;]R2)*. (4.21)

The first term on the right hand side of (4.21) is finite and vanishes as ί->0. The

second term is an infinite sum which converges for ί<(|/2e||ι;||1 2 ]R2)~1 andvanishes as tN as ί->0.

For the second term in (4.16) we use Holder's inequality to write

(eth-l-th) ^v

~ R2

" 2 . (4.22)

ϋsί eu°

The function (ev—l) for i eH is square integrable on L2(IR2,6?x). This followsfrom the bound

J (ev-\)2^ j \e2v-2v-l\+2 | |βϋ-ί;-l| (4.23)R2 R2 1R2

which is finite due to Lemma 4.6. We remark that J \eth—l — th\2 is order t4.

286 C. H. Taubes

For, given any ueH we have

oo I

Σ Z M| OΛ* (4.24)

The last step follows from the Monotone Convergence Theorem and the fact thatthe sum has strictly positive terms. Using Lemma 4.8 to bound INIS^.^ andchoosing an integer N so that inequality (4.20) holds we have

f \pu—\ _ 7 / / |2<7 V yJ \e i u\ ^z 2, L (-R2 n = 4 £ = 0 V"

+ 2 f; (2]/2eM| l ι2;R2)". (4.25)n = JV + 1

Setting u = th, we see that for sufficiently small ί, the right hand side of (4.25) isfinite and order ί4. This implies that as f->0 the right hand side of (4.22) vanishes,proving Lemma 4.7.

To prove the remaining assertions of Proposition 4.2 notice that

\a'(υ;h)\^ f \dμυdμh\+ f |Λ[|l-^o-^l+ ί W\ev-i\ . (4.26)R2 R2 R2

We use Holder's inequality to bound \a'(v, h)\ by

(4.27)U2 / J

The right hand side of (4,27) is finite for any veH. This completes the proof ofProposition 4.2.

Proof of Proposition 4.3. To prove that the functional a(v) is strictly convex wemust show that

Φ»ι +(1 -y)v2)^ya(vί) + (l-γ)a(v2) . (4.28)

For all vl9 v2εH; γe(0, 1) with equality only when vί = v2. This result follows fromthe strict convexity of the functional J d vdμv on H and the strict convexity of theexponential function. We leave the details to the reader.

V. The Existence of Weak Solutions

To prove the existence assertion of Theorem Γ it is sufficient to show that thefunctional a(v) satisfies the conditions set forth in Proposition 3.2. Gateauxdifferentiability of a(v) was proven in Proposition 4.2. Weak lower semi-continuityof the functional is a result of Corollary 4.4. It remains for us to show that thereexists a positive number R such that Eq. (3.8) is obeyed by <grad a(v\ vy for all vwith norm equal to R. The existence of such an R follows from the followingestimate:

Vortex Solutions to the Landau-Ginzburg Equations 287

Proposition 5.1. There exists constants α>0, b and /c>0 such that for all veH

-> °

The proof of Proposition 5.1 rests on the following properties of the functionsu0 and g0 :

Lemma 5.2. Let u0 and g0 be defined by Eq. (3.2) and (3.4) respectively. Let λ>4n.Then

a) There exists a constant cί>0 such that for all xeIR2

(5.2)

b) For all xeIR2

l-g0(x)-euo(x)>Q. (5.3)

Proof of Lemma 5.2. For part (a) we note that for λ > 4n

0o^ j<L (5.4)

For part (b) we have

(5 5)fc=l \{A — Uk) -ΓΛ) k=1

Define γ = 4n/λ < 1 and

With this notation we rewrite (5.5) as

n V "

k=l nfc=l

Each Zfc is less than 1 so that

Using this inequality in (5.7) gives finally

. 1 Λ ? A o~ Σ ^ ^ i - Γ Σ^<i (5-9)

n k=l n fe=l Π fc=l

Inequality (5.9) proves part (b) of the lemma.

Proqf o/ Proposition 5.i. Recall that the functional a'(v\υ) is given by

af(υ;υ)= f [3^5μt; + ϋ(^β + w 1 +^0)] . (5.10)IR2

We consider the expression v(euo + v— 1 +g0) pointwise.

288 C. H. Taubes

Case i. If φc) 0 then

It is simple to show that for any xeIR2,

<?*-l-x>0 (5.12)

with equality only at x = 0, so the third term in (5.11) is positive. Completing thesquare in (5.11) to eliminate the linear term we have for φc)g:0 and any βe(0, 1)the inequality

g0)2 . (5.13)0 0 0 .

U ~ P)

Case 2. If φc)^0 we have

v(e»° + v-l+g0) = \v\(l-g0-e"°) + \v\e»°(l-e-W). (5.14)

To continue further, notice that for x^O

l-e- (5.15)

with equality only when x = Q. Inequality (5.15) implies that

(5.16)

The last inequality in (5.16) results from the inequalities (a) and (b) of Lemma 5.2.Because β, c1 and (l + M)"1 are smaller than 1, we have proved that for any

βe(0, 1) and any veH

a'(v,v)^ J dμvδμv+ - _ - ί (u0 + 9of . (5.17)R2 \ l+\υ\ I (I-P) R2

The function (UQ + gQ)2 is integrable. We choose β= 1/2 and define

In order to continue we prove

Lemma 5.3. For any veH

Proof of Lemma 5.3. We use Holder's inequality to write

Vortex Solutions to the Landau-Ginzburg Equations 289

Inequality (5.19) is true if v = 0 so we may assume in (5.20) that i φO. Squaringboth sides of (5.20) and dividing by (IMIo52;]R2 + IMIo.s π^) yields the result.

Given a υeH, we define the number σe(0, 1) by

(5.21)ί dμvdμv = σ \\v\\l t2;K2 .

1R2

From Lemma 4.9 we have the inequality

Nθ,3;R2^Ml,2;R2> (5-22)

where /c = 8 ]/2 27. These definitions and Lemma 5.3 give

- " l ' -^. (5.23)

Finally, if we set ot = 3/4βcΐ we have for any veH the result:

This completes the proof of the existence and uniqueness of a weak solution toEqs. (3.6a) and (3.6b).

The functions u0 and g0 as defined in Eqs. (3.2) and (3.4) depend not only onthe points (α l5...,απ)eIR2 x . . .xlR 2 but also on the parameter λ. Our estimatesrequired λ>4n. Any two values, λί,λ2>0 of λ give the same solution. For let vt

satisfy (3.6a) and (3.6 b) withn / 1

r\ V^ ι / -i , ^iuo(1)=" Σ l n( 1 + 7 IΓ^

(5.25)

We note that u0(l) — uQ(2) is in if. By uniqueness then we have

. (5.26)

VI. Properties of the Solution

We now prove the regularity of the weak solution obtained in Sect. V. The Sobolevspace Pf^IR2) is defined as the completion of C^(IR2) in the norm

II <9αι 3α2 I IIML,p= Σ fc^feH (6'1}

α ι + α 2 ^ m ll17-^! UΛl2 Hθ,p;IR2

with α 1 ?α 2 nonnegative integers.We restate the final assertion of Theorem Γ in a proposition :

290 C. H. Taubes

Proposition 6.1. The unique weak solution, VQ to Eqs. (3.6a) and (3.6b) in H is realanalytic in IR2. D

Proof of Proposition 6Λ. Since u0 is a weak solution of (3.6a) and (3.6b),zlί;0eL2(ΪR2).

Thus v0 is in the Sobolev space PF2'2(IR2). By the Sobolev Imbedding Theorem([9], p. 97), ι;0eC°(lR2). Repeating this argument for derivatives of ι?0 we obtainve Cfe(IR2) for k = 0,1,.... By a standard theorem ([10], pp. 170-180) we have v0 isreal analytic in IR2.

Appendix

The purpose of this appendix is to prove the following proposition.

Proposition Al.l Let A and φ be respectively a C°° connection and Higgs field onIR2 which satisfy the first order Ginzburg- Landau equations. Then {xeJR-2\φ(x) = Q}is discrete. D

Proof of Proposition Al.l. Our plan is to assume the converse and show that acontradiction results. Therefore assume that there exists A and φ satisfying theconditions of Proposition Al.l with {xeIR2|φ(x) = 0} not discrete. Denote by Z thezero set of φ. If Z is not discrete then there exists a Jordan arc, y (for definitions,see e.g. Ahlfors [11], p. 69), in Z. Given any open set U intersecting γ there existsan open set Vc U such that y divides V into two nonempty sets F+ and F_ suchthat V+ n V_ — y and yn V+ is open in y. Because Z is the zero set of the C°° function\φ\2 on IR2, it is possible to choose a Jordan arc y CZ and an open set V as abovesuch that

By taking a smaller open set if necessary we may assume that V is simplyconnected.

Select an open set WC V such that y divides HP into two nonempty sets W+_andW_ with W+ simply connected; its closure W+ compact; its boundary dW+ aJordan curve and γnW+ open in y.

In the interior of F+, \n\φ\2 = u+ is a C°° function which satisfies

By assumption, \φ\2 is C°° in V. Standard arguments (see ej*., Lions and Magenes[12]) imply that there exists a unique C°° function h on W+ satisfying

- = n n t +

with _ (A1.3)ft = 0 on dW+ .

In the interior of W we have

Vortex Solutions to the Landau-Ginzburg Equations 291

Hence, in the interior of W+

with u a harmonic function in int W+. Since |φ|2=0 on 7 and is continuous in Vand ft|y = 0 it follows that for fixed ye 7

lim zφc)-» — oo. (A1.6)|x-y|-*0

xeintW+

Thus we are led to consider the possible behavior on the boundary, dW+9 of afunction u harmonic in int W+ . We note that it is sufficient to study this questionfor W+ the unit disc in 1R2 and 7 a subset of the unit circle. This follows from thefollowing two fundamental theorems on conformal mappings.

Theorem A1.2 (Riemann Mapping Theorem). The interior of any simply-connecteddomain R with more than one frontier-point can be represented on the interior of theunit circle by means of a one to one conformal transformation. D

Theorem A1.3. // one Jordan domain is transformed conformally into another, thenthe transformation is one-one and continuous in the closed domain, and the twofrontiers are described in the same sense by moving a point on one and thecorresponding point on the other. D

For proofs and discussions of these two theorems, see, e.g., Caratheodory [13],pp. 70-86.

Without loss of generality we take for ΘQ > 0

Here we are representing the unit disc, D, in IR2 as the set of complex numbers withmodulus less than or equal to one.

Lemma A1.4. Given rc>0, u(x) as above and W+=D, there exists 0<ρn<l suchthat

L) The set {θe[ΰ,2π~]\u(ρneiθ)< — n} has Lebesgue measure Θn>θ0 .

Proof of Lemma Ai.4. Statement L) follows from (A1.6) and the continuity of u inint D.

Let {ρw}*- i be a sequence of positive numbers which satisfy the conditions andstatement L) of Lemma A 1,4, with the added proviso that ρn < ρn + ί for n = 1, . . . oo.

We have l imρ =1.n-»oo *n

Since u is harmonic in int D,

(For a proof of this statement, see e.g., Ahlfors [11], p. 164.) w(0) is a finite constantindependent of n. Equation (A1.7) implies that

supβe[0,2π]

292 C. H. Taubes

Let znε{\z\ = ρn} be such that

sup u(ρneiθ) = u(zn) . (A1.9)

θe[0,2π]

The compactness of the interval [0,2π] insures that zn exists. The set {zn}™= x form asequence in D and hence have a limit point z^ e D. Since u is harmonic, z^ e dD andu(zao)= + 00. Applying the inverse mapping of D into FP+ we see that thiscontradicts that \φ\2 = Qxp{u + h} is continuous in the set V. Q.E.D.

Acknowledgements. We wish to thank Professor A. Jaffe and Thordur Jonsson for their suggestionsand critical comments.

References

1. Ginzburg, V.L., Landau, L.D.: Zh. Eksp. Teor. Fiz. 20, 1064 (1950)2. Nielsen, H.B., Olesen, P.: Nucl. Phys. B61, 45 (1973)3. deVega, H.J., Schnaposnik, F.A.: Phys. Rev. D14, 1100 (1976)4. Weinberg, E.: Phys. Rev. D 19, 3008 (1979)5. Jacobs. L, Rebbi, C.: Phys. Rev. B 19, 4486 (1979)6. BogomoPnyi, E.B.: Sov. J. of Nucl. Phys. 24, 449 (1976)7. Witten, E.: Phys. Rev. Lett. 38, 121 (1977)8. Vainberg, M.M.: Variational method and method of monotone ϊ theory of

nonlinear equations. New York: John Wiley 19739. Adams, R.: Sobolev spaces. New York: Academic Press 1975

10. Morrey, C.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York:Springer 1966

11. Ahlfors, L.V.: Complex analysis. New York: McGraw-Hill 196612. Lions, J., Magenes, E.: Nonhomogeneous boundary value problems and applications. Berlin,

Heidelberg, New York: Springer 197213. Caratheodory, C.: Conformal representation. Cambridge: Cambridge University Press 1963

Communicated by A. Jaffe

Received June 21, 1979; in revised form November 6, 1979